Following the optimization technique described in [1]}, approximating the expectation \(\mathbb {E}_{U\sim \mathcal {N}(\theta ,\kappa _s^2\mathrm {I}_{\mathrm {N}_p}\times \mathrm {I}_{\mathrm {N}_p})}\)  by averaging several
Monte-Carlo samples of \(U\) , the optimal posterior distribution can be obtained by evaluating the gradient of
(REF )
with respect to \((\mathbf {\mu }_i, \mathbf {\sigma }_i^2) \)  as described in Section 4.4 of [1]}.
